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Properties

Label 20.4.65846307453...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{28}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $49.08$
Ramified primes $2, 5, 3469$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15380, 55560, -65940, -76360, 424256, -682180, 663196, -416468, 191596, -111748, 62318, -23992, 11592, -4464, 1414, -502, 130, -60, 12, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 12*x^18 - 60*x^17 + 130*x^16 - 502*x^15 + 1414*x^14 - 4464*x^13 + 11592*x^12 - 23992*x^11 + 62318*x^10 - 111748*x^9 + 191596*x^8 - 416468*x^7 + 663196*x^6 - 682180*x^5 + 424256*x^4 - 76360*x^3 - 65940*x^2 + 55560*x - 15380)
 
gp: K = bnfinit(x^20 - 2*x^19 + 12*x^18 - 60*x^17 + 130*x^16 - 502*x^15 + 1414*x^14 - 4464*x^13 + 11592*x^12 - 23992*x^11 + 62318*x^10 - 111748*x^9 + 191596*x^8 - 416468*x^7 + 663196*x^6 - 682180*x^5 + 424256*x^4 - 76360*x^3 - 65940*x^2 + 55560*x - 15380, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 12 x^{18} - 60 x^{17} + 130 x^{16} - 502 x^{15} + 1414 x^{14} - 4464 x^{13} + 11592 x^{12} - 23992 x^{11} + 62318 x^{10} - 111748 x^{9} + 191596 x^{8} - 416468 x^{7} + 663196 x^{6} - 682180 x^{5} + 424256 x^{4} - 76360 x^{3} - 65940 x^{2} + 55560 x - 15380 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6584630745307392888012800000000000=2^{28}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.08$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 3469$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{2} a^{18}$, $\frac{1}{2205928699690524666780232578692937554554161988675872842} a^{19} - \frac{311617223111364879121328193665362490860023137455212593}{2205928699690524666780232578692937554554161988675872842} a^{18} + \frac{237976501228341771802373993509587479605072206320625825}{1102964349845262333390116289346468777277080994337936421} a^{17} + \frac{477901924723394114643115799495569094175697327750784415}{2205928699690524666780232578692937554554161988675872842} a^{16} - \frac{132274631323984414866113644065483257580398919180447153}{2205928699690524666780232578692937554554161988675872842} a^{15} - \frac{297895084239252439082185487997599709594236368593861491}{2205928699690524666780232578692937554554161988675872842} a^{14} + \frac{240300551524361957655995506891083767439302581236096807}{2205928699690524666780232578692937554554161988675872842} a^{13} + \frac{192028345683460663950016823834702611288304890094822757}{1102964349845262333390116289346468777277080994337936421} a^{12} + \frac{164247100088874492045084594771878286925224067418384399}{1102964349845262333390116289346468777277080994337936421} a^{11} - \frac{39246764819825467639443347388683469931444537879921464}{1102964349845262333390116289346468777277080994337936421} a^{10} + \frac{42679631196897272568668827409471254640999863294785324}{1102964349845262333390116289346468777277080994337936421} a^{9} - \frac{141245713950781936784008998790204408701992903908494972}{1102964349845262333390116289346468777277080994337936421} a^{8} + \frac{421462944667153773026805544194653878652678867304774996}{1102964349845262333390116289346468777277080994337936421} a^{7} + \frac{204240118398241436528685499967161114274940214132540442}{1102964349845262333390116289346468777277080994337936421} a^{6} + \frac{14976659816988321541878170743686389139460106894925670}{35579495156298784948068267398273186363776806268965691} a^{5} - \frac{32163905949220207912310108957783581549208806347009877}{1102964349845262333390116289346468777277080994337936421} a^{4} + \frac{482931346882546312138220137492588715619791187114013419}{1102964349845262333390116289346468777277080994337936421} a^{3} + \frac{394221957526322226147769338671446140622340710803531827}{1102964349845262333390116289346468777277080994337936421} a^{2} - \frac{375830489595551246294947928238898255237384828383612896}{1102964349845262333390116289346468777277080994337936421} a + \frac{64701562379854394728844027294886463630386231847075813}{1102964349845262333390116289346468777277080994337936421}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 591619411.361 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed